Dual Integral Equations Research Articles

Diffusiophoretic propulsion of an isotropic active colloidal particle near a finite-sized disk embedded in a planar fluid–fluid interface

Breaking spatial symmetry is an essential requirement for phoretic active particles to swim at low Reynolds number. This fundamental prerequisite for swimming at the micro scale is fulfilled either by chemical patterning of the surface of active particles or alternatively by exploiting geometrical asymmetries to induce chemical gradients and achieve self-propulsion. In the present paper, a far-field analytical model is employed to quantify the leading-order contribution to the induced phoretic velocity of a chemically homogeneous isotropic active colloid near a finite-sized disk of circular shape resting on an interface separating two immiscible viscous incompressible Newtonian fluids. To this aim, the solution of the phoretic problem is formulated as a mixed-boundary-value problem that is subsequently transformed into a system of dual integral equations on the inner and outer domains. Depending on the ratio of different involved viscosities and solute solubilities, the sign of phoretic mobility and chemical activity, as well as the ratio of particle–interface distance to the radius of the disk, the isotropic active particle is found to be repelled from the interface, be attracted to it, or reach a stable hovering state and remain immobile near the interface. Our results may prove useful in controlling and guiding the motion of self-propelled phoretic active particles near aqueous interfaces.

Solving Mixed Boundary Value Problems using Fourier Transform and Dual Integral Equations Method

The research considered the solution of the system of dual integral equations involving Fourier transform occurring in mixed boundary value problems for Laplace’s equation with mixed Dirichlet-Neumann boundary conditions. The dual integral equations involving trigonometric function kernel have been solved using Fourier sine transforms. Graphical solutions with the help of Mathematica were used to demonstrate the effectiveness of this method.
 Keywords: Boundary value problem; Laplace’s equation; Mixed boundary value problems; Fourier transform; Dual integral equations.

A Griffith Crack at the Interface of an Isotropic and Orthotropic Half Space Bonded Together

In the past 53 years, many efforts have been contributed to develop and demonstrate the properties of reinforced composite materials. The ever-increasing use of composite materials through engineering structures needs the proper analysis of the mechanical response of these structures. In the proposed work, we have an exact form of Stress components and Displacement components to a Griffith crack at the interface of an Isotropic and Orthotropic half-space bounded together. The expression was evaluated in the vicinity of crack tips by using Fourier transform method but here these components have been evaluated with the help of Fredholm integral equations and then reduce to the coupled Fredholm integral equations. In this paper, we use the problem of Lowengrub and Sneddon and reduce it to dual integral equations. Solution of these equations through the use of the method of Srivastava and Lowengrub is reduced to coupled Fredholm integral equation. Further reduces the problem to decoupled Fredholm integral equation of 2nd kind. We get the solution of dual integral equations and the problem is reduced to coupled Fredholm integral equation. We find the solution of the Fredholm integral equation and reduce it to decoupled Fredholm integral equation of 2nd kind. The Physical interest in fracture design criterion is due to Stress and crack opening Displacement components. In the end, we can easily calculate the Stress components and Displacement components in the exact form.

Axisymmetric Problem for Indentation of Multi-Layer Poroelastic Body

We considered the axisymmetric indentation of poroelastic multi-coating bonded to poroelastic substrate indented by a rigid flat-ended cylindrical indenter. The approach that expressed poroelastic solutions in term of two displacement functions and the transfer matrix method which can formulate the effect of multi-layer composite were applied in this study. Furthermore, we adopted the analytic method that using Laplace-Hankel transform and expressing the normal contact stress at the surface layer as an appropriate series with a Chebyshev orthogonal polynomial to reduce dual integral equations to an infinite system of simultaneous equations in Laplace transform domain. The numerical results of time-dependent axial displacement of rigid cylindrical indenter were presented by performing numerical inverse Laplace transform. The results of axial displacement for one-layer bonded to rigid impervious substrate given by this study were good agreement with previously reported results.

Horizontal vibration of rigid strip footings on poroelastic half-space

This paper presents an analytical method to determine the response of rigid strip footings to harmonic horizontal loads, while considering the foundation soil as poroelastic, two-phase medium. The derived solution is the first of its kind that allows considering in a robust manner the coupled effects of soil deformation and flow on the dynamic response of strip footings to horizontal loads. The strategy for treating this complex problem analytically comprises transforming the mixed boundary conditions to a pair of dual integral equations that provide the contact stresses at the soil-footing interface, which are solved by means of Jacobi orthogonal polynomials. Presentation of the solution is followed by a discussion on how soil flow parameters affect the dynamic response of strip footings to horizontal loads, and the importance of considering the foundation soil as two-phase medium.

Study of scattering from a PEC strip placed at the planar interface of NID dielectric and dielectric-magnetic medium using the KP method

Mathematical expressions for electromagnetic scattering from a Perfect electric conductor strip positioned at the planar non-integer dimension (NID) dielectric- magnetic interface are determined by using the Kobayashi potential (KP) method. The Lorentz-Drude model is used to include dispersion and losses in both permeability and permittivity of the dielectric- magnetic medium. Based on the KP method, the unknown scattered fields are presumed in terms of the spectrum with an undetermined weighting function. By imposing the appropriate boundary conditions, the dual integral equations for undetermined weighting functions obtained, the integrals are further solved by utilizing the property of Weber-Schafheitilin integrals. Then, using the orthogonal properties of Jacobi polynomials, the set of matrix equations satisfying the WSIs along with edge boundary conditions are obtained. This set of equations is solved numerically to find the undetermined expansion coefficients. Far-field expressions are studied to note the impact of the NID parameter on the scattered field. Based on the operating frequency, dielectric-magnetic medium behaves as double positive/mu negative/double negative/epsilon negative. It is studied that how the type of medium for lower half-space along with the presence of NID medium in upper half-space affects scattering characteristics of the strip.

An Elastic Half Space with a Moving Punch

The dynamic problem of a punch with rounded tips moving in an elastic half-space in a fixed direction has been considered. The static problem of determining stress component under the contact region of a punch has also been solved. Fourier integral transform has been employed to reduce the problems in solving dual integral equations. These integral equations have been solved using Cooke’s [1] result (1970) to obtain the stress component. Finally, exact expressions for stress components under the punch and the normal displacement component in the region outside the punch have been derived. Numerical results for stress intensity factor at the punch end and torque applied over the contact region have been presented in the form of graph.

Mechanics of multilayered or functionally graded material pressed onto an imperfect rigid base with a circular opening by localized surface loading

Loaded materials constrained by imperfect boundary can be found in many engineering scenarios, including blanking process, contact pressure sensor and soil/rock foundation subjected to underground mining. This paper examines the surface loading problem of a multilayered or functionally graded material (FGM) resting on an imperfect rigid smooth base weakened by a circular opening. The surface loading is axisymmetric and coaxial with the circular opening. This mixed boundary value problem is analogous to a Mode-I penny-shaped crack in multilayered elastic medium. With the aid of the GKS-based method for N-layered elasticity, the problem is formulated as a system of dual integral equations and then reduced to a Fredholm integral equation of the second kind. Solutions of the SIF, COD and full elastic field are expressed in terms of the solution of the integral equation. Numerical results obtained from the present solutions are compared with the those from finite element method. Using the new solutions, numerical studies are also conducted for four examples, including a homogenous layer, a multilayered material, a material with FGM coating and an asphalt pavement structure with FGM properties. These numerical results demonstrate that (1) a small size of surface loading can result in high SIF value at the crack tip; (2) placing the hard layer as the top surface is most effective for reducing ground settlement; (3) the gradation in Poisson’s ratio can have non-negligible effect on the SIF for thin FGM coating; (4) effect of the circular opening to the ground settlement can be significant if its radius is comparable to the total thickness of the upper layers.

Dynamic impedances of ring disks buried in arbitrary depths

This paper addresses the vertical, horizontal and rocking vibrations of a rigid ring disk, with especially small radial wall-thicknesses, embedded in any depth of a fully saturated poroelastic half-space. The mixed boundary value problem is expressed in terms of dual integral equations that are solved semi-analytically with all boundary conditions satisfied. Numerical results of quasi-static stiffness, dynamic impedances (including dynamic stiffness and radiation damping ratio) are obtained. The influence of Poisson’s ratio, soil permeability, disk embedded depth and excitation frequency on the stiffness and damping ratio of the ring disk is illustrated. It is found that for a ring disk, although its thickness-to-radius ratio is only about 2%-4%, its vertical, horizontal and rocking impedances are around 70% of the corresponding values of a circular disk. Besides, the vertical, horizontal and rocking stiffness of a deeply embedded ring disk is all about 2.0 times that of the surface ring disk. Simplified equations for the initial base stiffness of pipe piles/caissons are obtained. The results can be combined with Winkler-based models to address thin-walled pile/caisson-soil interaction problems in the field of offshore wind turbines or bridges.

On the receding contact between a graded and a homogeneous layer due to a flat-ended indenter

This paper solves the frictionless receding contact problem between a graded and a homogeneous elastic layer due to a flat-ended rigid indenter. Although its Poisson’s ratio is kept as a constant, the shear modulus in the graded layer is assumed to exponentially vary along the thickness direction. The primary goal of this study is to investigate the functional dependence of both contact pressures and the extent of receding contact on the mechanical and geometric properties. For verification and validation purposes, both theoretical analysis and finite element modelings are conducted. In the analytical formulation, governing equations and boundary conditions of the double contact problem are converted into dual singular integral equations of Cauchy type with the help of Fourier integral transforms. In view of the drastically different singularity behavior of the stationary and receding contact pressures, Gauss–Chebyshev quadratures and collocations of both the first and the second kinds have to be jointly used to transform the dual singular integral equations into an algebraic system. As the resultant algebraic equations are nonlinear with respect to the extent of receding contact, an iterative algorithm based on the method of steepest descent is further developed. The semianalytical results are extensively verified and validated with those obtained from the graded finite element method, whose implementation details are also given for easy reference. Results from both approaches reveal that the property gradation, indenter width, and thickness ratio all play significant roles in the determination of both contact pressures and the receding contact extent. An appropriate combination of these parameters is able to tailor the double contact properties as desired.

Steady azimuthal flow field induced by a rotating sphere near a rigid disk or inside a gap between two coaxially positioned rigid disks

Geometric confinements play an important role in many physical and biological processes and significantly affect the rheology and behavior of colloidal suspensions at low Reynolds numbers. On the basis of the linear Stokes equations, we investigate theoretically and computationally the viscous azimuthal flow induced by the slow rotation of a small spherical particle located in the vicinity of a rigid no-slip disk or inside a gap between two coaxially positioned rigid no-slip disks of the same radius. We formulate the solution of the hydrodynamic problem as a mixed-boundary-value problem in the whole fluid domain, which we subsequently transform into a system of dual integral equations. Near a stationary disk, we show that the resulting integral equation can be reduced into an elementary Abel integral equation that admits a unique analytical solution. Between two coaxially positioned stationary disks, we demonstrate that the flow problem can be transformed into a system of two Fredholm integral equations of the first kind. The latter are solved by means of numerical approaches. Using our solution, we further investigate the effect of the disks on the slow rotational motion of a colloidal particle and provide expressions of the hydrodynamic mobility as a function of the system geometry. We compare our results with corresponding finite-element simulations and observe very good agreement.

Dual Integral Equations with a Trigonometric Kernel

Dual Integral Equations with a Trigonometric Kernel

Electromagnetic Scattering and Emission From Large Rough Surfaces With Multiple Elevations Using the MLSD-SMCG Method

Electromagnetic scattering and emission from 1-D rough surfaces with multiple elevations are studied using full-wave simulations. Both the root-mean-square (rms) heights and the surface length are large compared to the wavelength. A novel multilevel steepest decent-sparse matrix canonical grid (MLSD-SMCG) method is proposed to address limitations in the original SMCG. The uniform Nystrom method and neighborhood impedance boundary condition (NIBC) are also incorporated in solving the dual surface integral equations (SIEs) of the method of moments (MoM). Simulation results are illustrated at L-band for soil and ocean surfaces. The surface rms heights and lengths are up to 1.43 and 243.8 m corresponding to 6 and 1024 wavelengths at 1.26 GHz, respectively. For ocean surfaces, the wind speeds up to 20 m/s are considered, and the entire spectrum is included to capture all relevant surface length scales. Numerical results indicate the proposed approach is computationally efficient and accurate. Energy conservation checks in simulations are at $10^{-4}$ for ocean scattering and emission. Also, the effects of wind-driven roughness on ocean emissivity are further investigated using the proposed approach in terms of wind speed and observation angle for both polarizations.

Dynamic fracture behavior for functionally graded piezoelectric bi-materials with interfacial cracks near a circular hole

This work aims to develop an effective method for evaluating the stress concentration at the tip of permeable interfacial cracks near a circular hole in functionally graded piezoelectric bi-materials. The structure is loaded by anti-plane incident SH-wave and all material parameters are assumed to obey exponential variations. Fracture analysis is performed by the Green function method, which is used to solve the boundary conditions problem. The mechanical model of the cracks is constructed by crack-conjunction and crack-deviation techniques so that the crack problem is simplified as solving a series of the first kind of Fredholm’s integral equations, from which the dynamic stress intensity factors (DSIFs) at the inner and the outer crack tips can be derived. The validity of the present method is verified by comparison with references. Numerical cases reveal parametric dependence of DSIFs on the geometry of circular holes and cracks, the characteristics of the incident waves and the inhomogeneity of materials. The method proposed in this paper can circumvent the limitations of using dual integral equations to deal with asymmetric defects and opens up a new way for the research on fracture problems in functionally graded piezoelectric materials with more complex defects.

Dynamic impedances and load carrying mechanism for skirted foundations

Dynamic impedances of foundations include dynamic stiffness and damping which have important effect on the internal forces in the structure. In some cases, such as offshore wind turbines, the influence of the foundation impedances on the system's natural frequency and overall damping could potentially have a significant effect on the fatigue life of the structure. The vertical, horizontal and rocking impedances of a skirted foundation (also termed bucket foundation in offshore wind industry) embedded in a fully saturated poroelastic seabed are addressed in this paper. The vertical impedance is most relevant for jacket foundations supported on three or four bucket foundations, while horizontal and rocking impedances are applicable for mono-bucket foundations. The dynamic vibration problems are solved semi-analytically with the help of dual integral equations and Green's functions. Numerical results for dynamic impedances are obtained; damping ratio are also obtained to show the importance of radiation damping for bucket foundations, even at very small excitation frequencies. The influence of length-to-radius ratio, Poisson's ratio, permeability of soil, excitation frequency and thickness-to-radius ratio on the impedances are also studied. Besides, the dynamic load sharing among the top plate, bucket shaft and bucket tip is obtained for vertical load, horizontal load and moment to shed light on the carrying mechanism of bucket foundation at dynamic working loads. It is found that for a rigid bucket foundation, even when the length-to-radius ratio is small (e.g. l/a = 1.0), most of the loads are carried by the shaft, while the top plate and tip of the bucket take only a small portion of the loads. The results of this study will be helpful for understanding the load-carrying mechanism of offshore bucket foundations for normal operation conditions.

An analytical approach to rotor blade modulation

Scattering of electromagnetic plane waves normally incident on a flat perfect electric conductor strip in arbitrary rotational motion in its own plane is investigated in the context of Hertzian Electrodynamics. This canonical problem is an analytical-asymptotic approach that reveals the parametric dependence of interference of electromagnetic waves with rotating blades, a subject which is known as rotor blade modulation. Scattering by a stationary perfect electric conductor strip under TE and TM incidences stand as two sub-problems that are incorporated in the main formulation. These sub-problems are formulated by Wiener–Hopf Technique and solved using a spectral iteration method that converges rapidly in the solution of corresponding coupled dual Fredholm type integral equations in case of electrically wide strips. Numerical implementations that simulate electromagnetic scattering by helicopters blades and wind turbines are provided and discussed.

Scattering from a PEMC strip deeply buried beneath a dielectric slab using Kobayashi potential method

Kobayashi potential method is applied to get the scattering behavior by a perfect electromagnetic conducting strip deeply buried underneath a flat dielectric slab. Using the two-dimensional solution of homogeneous Helmholtz’s equation, Fourier transformed representation of scattered field is written in terms of the unknown characteristic functions. Application of the boundary conditions on tangential components of electromagnetic fields gives dual integral equations (DIEs) which through Weber–Schafheitlin’s discontinuous integrals and Jacobi polynomials are used to satisfy the edge condition. The resulting DIEs are written in matrix equations, and those that involve infinite integrals are computed numerically. The saddle point method of integration is used to obtain an expression for the far-zone scattered field. To validate the results, obtained results are firstly compared with corresponding obtained through the physical optics method. The width of the strip, the thickness of the slab, and the angle of incidence of plane wave are taken as parameters to study their influence on the scattered field.

Interior-stress fields produced by a general axisymmetric punch

This work is a supplement to the work of Sneddon on axisymmetric Boussinesq problem in 1965 in which the distributions of interior-stress fields are derived here for a punch with general profile. A novel set of mathematical procedures is introduced to process the basic elastic solutions (obtained by the method of Hankel transform, which was pioneered by Sneddon) and the solution of the dual integral equations. These processes then enable us to not only derive the general relationship of indentation depth D and total load P that acts on the punch but also explicitly obtain the general analytical expressions of the stress fields beneath the surface of an isotropic elastic half-space. The usually known cases of punch profiles are reconsidered according to the general formulas derived in this study, and the deduced results are verified by comparing them with the classical results. Finally, these general formulas are also applied to evaluate the von Mises stresses for several new punch profiles.

On the frictional receding contact between a graded layer and an orthotropic substrate indented by a rigid flat-ended stamp

This paper solves the frictional receding contact between a functionally graded elastic layer and an orthotropic substrate under the indentation of a rigid flat-ended stamp. Both normal and tangential indentation forces, satisfying Coulomb's friction law, are considered. The shear modulus of the graded layer is assumed to vary exponentially along the transverse dimension, whereas the Poisson's ratio is fixed as a constant. Under the plane strain sliding condition, the governing equations and boundary conditions of the double contact problem are converted into dual singular integral equations with the help of Fourier integral transforms. Gauss-Jacobi quadratures of different kinds and an iterative algorithm are subsequently developed to determine the stationary contact pressure, the receding contact pressure as well as the asymmetric receding contact boundaries. In order to verify and validate the accuracy of semianalytical solutions, the graded finite element modelings are also implemented through defining material properties of the graded layer at the Gaussian integral points of all elements and communicating with the major solver in terms of a user-defined interface. Parametric studies primarily focus on the effects of frictional coefficients, stiffness orthotropy and gradation on the stationary and receding contact properties. The influence of indenter size and thickness ratio between the graded layer and the orthotropic substrate is also investigated. Reasonable agreements of contact pressures and receding contact lengths are identified between the solutions obtained from two independent approaches. Extensive parametric studies provide an effective means of tailoring the double contact properties as desired by the appropriate selection of governing parameters.

Magnetic field penetration through a circular aperture in a perfectly conducting plate excited by a coaxial loop

The transmission of an electromagnetic field produced by a current loop of finite radius through a coaxial circular aperture in a perfectly conducting plate is evaluated through a rapidly convergent formulation in an exact form. By applying the equivalence principle, the problem is first formulated in the Hankel transform domain, obtaining a set of dual integral equations in which the equivalent surface magnetic current density defined on the aperture is not known. The set of dual integral equations is regularised in a second-kind Fredholm integral equation by applying the Abel integral-transform technique. The solution is achieved by expanding the unknown in a set of orthogonal basis functions that correctly reproduce the behaviour of the equivalent magnetic current at the edge of the aperture. Finally, under particular assumptions, a low-frequency solution is extracted in a closed form. Numerical results are reported to validate the accuracy and efficiency of the proposed formulations.